Question

mhf4u1 quiz v - factor theorem
name: hossam emari
answer all questions in the space provided.

given the following polynomial:
i) ( f(x) = 2x^4 - x^3 - 9x^2 + 4x + 4
ii) ( g(x) = x^4 + 2x^3 + x^2 + 8x - 12

a) using the r.r.t. determine the possible rational roots.
b) determine all the possible root combinations (real or complex).
c) factor fully.
d) sketch the graph.
e) determine the intervals in which the ( f(x) < 0 )

Explanation:

Step1: Recall Rational Root Theorem (R.R.T.)

The Rational Root Theorem states that any possible rational root of a polynomial \( f(x) = a_nx^n + a_{n - 1}x^{n - 1}+\dots+a_1x + a_0\) (where \(a_n eq0\)) is of the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term \(a_0\) and \(q\) is a factor of the leading coefficient \(a_n\).

For \(f(x)=2x^4 - x^3 - 9x^2 + 4x + 4\):

- Leading coefficient \(a_n = 2\), factors of \(2\) (i.e., \(q\)): \(\pm1,\pm2\)
- Constant term \(a_0 = 4\), factors of \(4\) (i.e., \(p\)): \(\pm1,\pm2,\pm4\)
- Possible rational roots: \(\frac{p}{q}=\pm1,\pm2,\pm4,\pm\frac{1}{2}\)

For \(g(x)=x^4 + 2x^3 + x^2 + 8x - 12\):

- Leading coefficient \(a_n = 1\), factors of \(1\) (i.e., \(q\)): \(\pm1\)
- Constant term \(a_0=- 12\), factors of \(-12\) (i.e., \(p\)): \(\pm1,\pm2,\pm3,\pm4,\pm6,\pm12\)
- Possible rational roots: \(\frac{p}{q}=\pm1,\pm2,\pm3,\pm4,\pm6,\pm12\)

Step2: Determine Root Combinations (Fundamental Theorem of Algebra)

The Fundamental Theorem of Algebra states that a polynomial of degree \(n\) has exactly \(n\) roots (real or complex), counting multiplicities. Also, complex roots come in conjugate pairs (if the polynomial has real coefficients, which both \(f(x)\) and \(g(x)\) do).

For \(f(x)\) (degree 4):

- Let the roots be \(r_1,r_2,r_3,r_4\). The possible combinations (considering real and complex roots):
- 4 real roots (could be with multiplicities, e.g., two pairs of real roots, or a root with multiplicity 2 and two other real roots, etc.)
- 2 real roots and 1 pair of complex conjugate roots
- 0 real roots and 2 pairs of complex conjugate roots

For \(g(x)\) (degree 4):

- Let the roots be \(s_1,s_2,s_3,s_4\). The possible combinations:
- 4 real roots
- 2 real roots and 1 pair of complex conjugate roots
- 0 real roots and 2 pairs of complex conjugate roots

Step3: Factor Fully (Using Rational Root Theorem and Polynomial Division/ Factoring)

For \(f(x)=2x^4 - x^3 - 9x^2 + 4x + 4\):

- Test possible rational roots. Let's test \(x = 1\): \(f(1)=2 - 1-9 + 4 + 4=0\). So \(x - 1\) is a factor.
- Divide \(f(x)\) by \(x - 1\) using polynomial long division or synthetic division.
- Using synthetic division for \(x = 1\):
\[ \begin{array}{r|rrrrr} 1 & 2 & -1 & -9 & 4 & 4\\ & & 2 & 1 & -8 & -4\\ \hline & 2 & 1 & -8 & -4 & 0\\ \end{array} \]
- So \(f(x)=(x - 1)(2x^3 + x^2 - 8x - 4)\)
- Now factor \(2x^3 + x^2 - 8x - 4\). Group terms: \((2x^3 + x^2)-(8x + 4)=x^2(2x + 1)-4(2x + 1)=(x^2 - 4)(2x + 1)=(x - 2)(x + 2)(2x + 1)\)
- Thus, \(f(x)=(x - 1)(x - 2)(x + 2)(2x + 1)\)

For \(g(x)=x^4 + 2x^3 + x^2 + 8x - 12\):

- Test possible rational roots. Let's test \(x = 1\): \(g(1)=1 + 2+1 + 8-12=0\). So \(x - 1\) is a factor.
- Divide \(g(x)\) by \(x - 1\) using synthetic division:
\[ \begin{array}{r|rrrrr} 1 & 1 & 2 & 1 & 8 & -12\\ & & 1 & 3 & 4 & 12\\ \hline & 1 & 3 & 4 & 12 & 0\\ \end{array} \]
- So \(g(x)=(x - 1)(x^3 + 3x^2 + 4x + 12)\)
- Factor \(x^3 + 3x^2 + 4x + 12\) by grouping: \((x^3 + 3x^2)+(4x + 12)=x^2(x + 3)+4(x + 3)=(x^2 + 4)(x + 3)\)
- Since \(x^2+4=(x - 2i)(x + 2i)\) (complex factors), the fully factored form (over complex numbers) is \(g(x)=(x - 1)(x + 3)(x - 2i)(x + 2i)\); over real numbers, it is \(g(x)=(x - 1)(x + 3)(x^2 + 4)\)

Step4: Sketch the Graph (For \(f(x)\))

- **End - behavior**: For \(f(x)=2x^4 - x^3 - 9x^2 + 4x + 4\), the leading term is \(2x^4\). As \(x ightarrow\pm\infty\), \(y = f(x) ightarrow+\infty\) (since the leading coefficient \(2>0\) and degree 4 is even).
- **x - intercepts**: From the factored form \(f(x)=(x - 1)(x - 2)(x + 2)(2x + 1)\), the x - intercepts are \(x = 1,x = 2,x=-2,x=-\frac{1}{2}\)
- **y - intercept**: \(f(0)=4\)
- **Test intervals between intercepts** to determine the sign of \(f(x)\) in those intervals. For example, for \(x < - 2\), pick \(x=-3\): \(f(-3)=2(81)-(-27)-9(9)+4(-3)+4=162 + 27-81-12 + 4=100>0\); for \(-2<x<-\frac{1}{2}\), pick \(x=-1\): \(f(-1)=2(1)-(-1)-9(1)+4(-1)+4=2 + 1-9-4 + 4=-6<0\); for \(-\frac{1}{2}<x<1\), pick \(x = 0\): \(f(0)=4>0\); for \(1<x<2\), pick \(x = 1.5\): \(f(1.5)=2(\frac{81}{16})-(\frac{27}{8})-9(\frac{9}{4})+4(\frac{3}{2})+4=\frac{81}{8}-\frac{27}{8}-\frac{81}{4}+6 + 4=\frac{54}{8}-\frac{162}{8}+10=\frac{54 - 162 + 80}{8}=\frac{-28}{8}<0\); for \(x>2\), pick \(x = 3\): \(f(3)=2(81)-27-9(9)+12 + 4=162-27 - 81+12 + 4=70>0\)
- Plot the intercepts, use the end - behavior and the sign of the function in intervals to sketch the graph (a "W" - shaped graph with turning points between the intercepts)

Step5: Determine Intervals where \(f(x)<0\) (For \(f(x)\))

From the sign analysis in Step4:
- The intervals where \(f(x)<0\) are \((-2,-\frac{1}{2})\) and \((1,2)\)

Answers:

a)
- For \(f(x)\): Possible rational roots are \(\boldsymbol{\pm1,\pm2,\pm4,\pm\frac{1}{2}}\)
- For \(g(x)\): Possible rational roots are \(\boldsymbol{\pm1,\pm2,\pm3,\pm4,\pm6,\pm12}\)

b)
- For \(f(x)\) (degree 4): Possible root combinations: 4 real roots; 2 real roots and 1 pair of complex conjugate roots; 0 real roots and 2 pairs of complex conjugate roots.
- For \(g(x)\) (degree 4): Possible root combinations: 4 real roots; 2 real roots and 1 pair of complex conjugate roots; 0 real roots and 2 pairs of complex conjugate roots.

c)
- \(f(x)=\boldsymbol{(x - 1)(x - 2)(x + 2)(2x + 1)}\)
- \(g(x)\) (over real numbers): \(\boldsymbol{(x - 1)(x + 3)(x^2 + 4)}\); (over complex numbers): \(\boldsymbol{(x - 1)(x + 3)(x - 2i)(x + 2i)}\)

d) (Sketch Description) The graph of \(f(x)\) has end - behavior \(y ightarrow+\infty\) as \(x ightarrow\pm\infty\), x - intercepts at \(x=-2,x =-\frac{1}{2},x = 1,x = 2\), y - intercept at \(y = 4\), and is negative in \((-2,-\frac{1}{2})\) and \((1,2)\), positive otherwise.

e) For \(f(x)\), the intervals where \(f(x)<0\) are \(\boldsymbol{(-2,-\frac{1}{2})\cup(1,2)}\)

Answer:

s:
a)

• For \(f(x)\): Possible rational roots are \(\boldsymbol{\pm1,\pm2,\pm4,\pm\frac{1}{2}}\)
• For \(g(x)\): Possible rational roots are \(\boldsymbol{\pm1,\pm2,\pm3,\pm4,\pm6,\pm12}\)

b)

• For \(f(x)\) (degree 4): Possible root combinations: 4 real roots; 2 real roots and 1 pair of complex conjugate roots; 0 real roots and 2 pairs of complex conjugate roots.
• For \(g(x)\) (degree 4): Possible root combinations: 4 real roots; 2 real roots and 1 pair of complex conjugate roots; 0 real roots and 2 pairs of complex conjugate roots.

c)

• \(f(x)=\boldsymbol{(x - 1)(x - 2)(x + 2)(2x + 1)}\)
• \(g(x)\) (over real numbers): \(\boldsymbol{(x - 1)(x + 3)(x^2 + 4)}\); (over complex numbers): \(\boldsymbol{(x - 1)(x + 3)(x - 2i)(x + 2i)}\)

d) (Sketch Description) The graph of \(f(x)\) has end - behavior \(y
ightarrow+\infty\) as \(x
ightarrow\pm\infty\), x - intercepts at \(x=-2,x =-\frac{1}{2},x = 1,x = 2\), y - intercept at \(y = 4\), and is negative in \((-2,-\frac{1}{2})\) and \((1,2)\), positive otherwise.

e) For \(f(x)\), the intervals where \(f(x)<0\) are \(\boldsymbol{(-2,-\frac{1}{2})\cup(1,2)}\)